IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES


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1 IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES Charles J. Corrado Department of Finance University of Missouri  Columbia Tie Su Department of Finance University of Miami  Coral Gables August 3, 1997 ABSTRACT The BlackScholes (1973) option pricing model is used to value a wide range of option contracts. However, the model often inconsistently prices deep inthemoney and deep outofthemoney options. Options professionals refer to this phenomenon as a volatility skew or smile. In this paper, we apply an extension of the BlackScholes model developed by Jarrow and Rudd (198) to an investigation of S&P 500 index option prices. We find that nonnormal skewness and kurtosis in optionimplied distributions of index returns contribute significantly to the phenomenon of volatility skews. Correspondence to Charles Corrado, 14 Middlebush Hall, University of Missouri, Columbia, MO 6511, (573) FAX
2 IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES The BlackScholes (1973) option pricing model is used to value a wide range of option contracts. However, the model often inconsistently prices deep inthemoney and deep outofthemoney options. Options professionals refer to this phenomenon as a volatility skew or smile. In this paper, we apply an extension of the BlackScholes model developed by Jarrow and Rudd (198) to an investigation of S&P 500 index option prices. We find that nonnormal skewness and kurtosis in optionimplied distributions of index returns contribute significantly to the phenomenon of volatility skews. The BlackScholes (1973) option pricing model provides the foundation of modern option pricing theory. In actual applications, however, the model has the known deficiency of often inconsistently pricing deep inthemoney and deep outofthemoney options. Options professionals refer to this phenomenon as a volatility skew or smile. A volatility skew is the anomalous pattern that results from calculating implied volatilities across a range of strike prices. Typically, the skew pattern is systematically related to the degree to which the options are in or outofthemoney. This phenomenon is not predicted by the BlackScholes model, since volatility is a property of the underlying instrument and the same implied volatility value should be observed across all options on the same instrument. The BlackScholes model assumes that stock prices are lognormally distributed, which in turn implies that stock logprices are normally distributed. Hull (1993) and Nattenburg (1994) point out that volatility skews are a consequence of empirical violations of the normality assumption. In this paper, we investigate volatility skew patterns embedded in S&P 500 index option prices. We adapt a method developed by Jarrow and Rudd (198) to extend the Black Scholes formula to account for nonnormal skewness and kurtosis in stock returns. This method fits the first four moments of a distribution to a pattern of empirically observed option prices. The mean of this distribution is determined by option pricing theory, but an estimation procedure is employed to yield implied values for the standard deviation, skewness and kurtosis of the distribution of stock index prices. 1
3 We organize this paper as follows. In the next section, we examine the first four moments of the historical distributions of monthly S&P 500 index returns. Next, we review Jarrow and Rudd s (198) development of a skewness and kurtosisadjusted BlackScholes option price formula and show how nonnormal skewness and kurtosis in stock return distributions give rise to volatility skews. We then describe the data sources used in our empirical analysis. In the subsequent empirical section, we assess the performance improvement of JarrowRudd s skewness and kurtosisadjusted extension to the BlackScholes model. Following this, we discuss the implications of the JarrowRudd model for hedging strategies. The final section summarizes and concludes the paper. [Exhibit 1 here.] I. NONNORMAL SKEWNESS AND KURTOSIS IN STOCK RETURNS It is widely known that stock returns do not always conform well to a normal distribution. As a simple examination, we separately compute the mean, standard deviation, and coefficients of skewness and kurtosis of monthly S&P 500 index returns in each of the seven decades from 196 through These are reported in Exhibit 1, where Panel A reports statistics based on arithmetic returns and Panel B reports statistics based on logrelative returns. Arithmetic returns are calculated as P t /P t11 and logrelative returns are calculated as log(p t /P t1 ), where P t denotes the index value observed at the end of month t. The returns series used to compute statistics reported in Exhibit 1 do not include dividends. We choose returns sans dividends because dividends paid out over the life of a Europeanstyle option do not accrue to the option holder. Thus Europeanstyle S&P 500 index option prices are properly determined by index returns that exclude any dividend payments. The BlackScholes model assumes that arithmetic returns are lognormally distributed, or equivalently, that logrelative returns are normally distributed. All normal distributions have a skewness coefficient of zero and a kurtosis coefficient of 3. All lognormal distributions are positively skewed with kurtosis always greater than 3 (Stuart and Ord, 1987). However, from Exhibit 1 we make two observations. First, reported coefficients of skewness and kurtosis show significant deviations from normality occurring in the first two decades ( and )
4 and the most recent decade ( ) of this 70year period. Statistical significance is assessed by noting that population skewness and kurtosis for a normal distribution are 0 and 3, respectively. Also, variances of sample coefficients of skewness and kurtosis from a normal population are 6/n and 4/n, respectively. For each decade, n = 10 months which is sufficiently large to invoke the central limit theorem and make the assumption that sample coefficients are normally distributed (Stuart and Ord, 1987, p.338). Thus, 95% confidence intervals for a test of index return normality are given by ± /10 = ± for sample skewness and 3 ± /10 = 3 ± for sample kurtosis. For statistics computed from logrelative returns, sample skewness and kurtosis values outside these confidence intervals indicate statistically significant departures from normality. For statistics obtained from arithmetic returns, a negative sample skewness value outside the appropriate confidence interval indicates a statistically significant departure from lognormality. Second, statistics reported for the decade are sensitive to the inclusion of the October 1987 return when the S&P 500 index fell by 1.76%. Including the October 1987 return yields logrelative skewness and kurtosis coefficients of and 11.9, respectively, which deviate significantly from a normal specification. By contrast, excluding the October 1987 return yields skewness and kurtosis coefficients of 0.0 and 4.05 which are not significant deviations from normality. The contrasting estimates of S&P 500 index return skewness and kurtosis in the decade raise an interesting empirical issue regarding the pricing of S&P 500 index options. Specifically, do postcrash option prices embody an ongoing market perception of the possibility of another market crash similar to that of October 1987? If postcrash option prices contain no memory of the crash, then the nearnormal skewness and kurtosis obtained by omitting the October 1987 return suggest that the BlackScholes model should be well specified. However if postcrash option prices remember the crash, then we expect to see nonnormal skewness and kurtosis in the optionimplied distribution of stock returns similar to the sample skewness and kurtosis obtained by including the October 1987 return. The JarrowRudd option pricing model provides a useful analytic tool to examine these contrasting hypotheses. 3
5 II. JARROWRUDD SKEWNESS AND KURTOSISADJUSTED MODEL Jarrow and Rudd (198) propose a method to value European style options when the underlying security price at option expiration follows a distribution F known only through its moments. They derive an option pricing formula from an Edgeworth series expansion of the security price distribution F about an approximating distribution A. While their analysis yields several variations, their simplest option pricing formula is the following expression for an option price. C( F) = C( A) e rt κ 3( F) κ 3( A) da( K) rt κ 4( F) κ 4( A) d a( K) + e 3! ds 4! ds t t + ε( K) (1) The left hand term C(F) above denotes a call option price based on the unknown price distribution F. The first right hand term C(A) is a call price based on a known distribution A, followed by adjustment terms based on the cumulants κ j (F) and κ j (A) of the distributions F and A, respectively, and derivatives of the density of A. The density of A is denoted by a(s t ), where S t is a random stock price at option expiration. These derivatives are evaluated at the strike price K. The remainder ε(k) continues the Edgeworth series with terms based on higher order cumulants and derivatives. Cumulants are similar to moments. In fact, the first cumulant is equal to the mean of a distribution and the second cumulant is equal to the variance. The JarrowRudd model uses third and fourth cumulants. The relationship between third and fourth cumulants and moments for a distribution F are: κ 3 (F) = µ 3 (F) and κ 4 (F) = µ 4 (F)  3µ (F), where µ is the squared variance and µ 3, µ 4 denote third and fourth central moments (Stuart and Ord, 1987, p.87). Thus the third cumulant is the same as the third central moment and the fourth cumulant is equal to the fourth central moment less three times the squared variance. Jarrow and Rudd (198) suggest that with a good choice for the approximating distribution A, higher order terms in the remainder ε(k) are likely to be negligible. In essence, the JarrowRudd model relaxes the strict distributional assumptions of the BlackScholes model without requiring an exact knowledge of the true underlying distribution. Because of its preeminence in option pricing theory and practice, JarrowRudd suggest the lognormal 4
6 distribution as a good approximating distribution. When the distribution A is lognormal, C(A) becomes the familiar BlackScholes call price formula. In a notation followed throughout this paper, the BlackScholes call price formula is stated below where S 0 is a current stock price, K is a strike price, r is an interest rate, t is the time until option expiration, and the parameter σ is the instantaneous variance of the security logprice. rt ( ) ( ) C( A) = S N d Ke N d 0 1 () d 1 = ( 0 ) + ( + σ ) log S / K r / t σ t d = d σ t 1 For the reader s reference, evaluating the lognormal density a(s t ) and its first two derivatives at the strike price K yields these expressions. a( K) = ( K t ) 1 σ π exp( d / ) da( K) a( K) d = ds Kσ t ( σ t ) t (3) ( d σ t) σ t( d σ t) d a( K) a( K) = ds K σ t t 1 The riskneutral valuation approach adopted by Jarrow and Rudd (198) implies equality of the first cumulants of F and A, i.e., κ 1 (F) = κ 1 (A) = S 0 e rt. This is equivalent to the equality of the means of F and A, since the first cumulant of a distribution is its mean. Also, the call price in equation (1) corresponds to JarrowRudd s first option price approximation method. This method selects an approximating distribution that equates second cumulants of F and A, i.e., κ (F) = κ (A). This is equivalent to the equality of the variances of F and A, since the second cumulant of a distribution is equal to its variance. Consequently, JarrowRudd show that when the distribution A is lognormal the volatility parameter σ is specified as a solution to the equality κ σ ( 1) ( F) = κ ( A) e t. 1 Dropping the remainder term ε(k), the JarrowRudd option price in equation (1) is conveniently restated as 5
7 C( F) = C( A) + λ Q + λ Q (5) where the terms λ j and Q j, j = 1,, above are defined as follows. σ t ( 1) rt 3 λ = γ ( F) γ ( A) Q = ( S e ) e / rt e 3! da( K) ds t (6.a) σ t ( 1) rt 4 λ = γ ( F) γ ( A) Q = ( S e ) e 4 0 e 4! rt d a( K) ds t (6.b) In equation (6), γ 1 (F) and γ 1 (A) are skewness coefficients for the distributions F and A, respectively. Similarly, γ (F) and γ (A) are excess kurtosis coefficients. Skewness and excess kurtosis coefficients are defined in terms of cumulants as follows [Stuart and Ord (1987, p.107)]. γ 1 κ 3( F) κ 4( F) ( F) = γ ( F) = (7) κ 3 / ( F) κ ( F) Coefficients of skewness and excess kurtosis for the lognormal distribution A are defined below, where the substitution q γ 1 σ e t = 1 is used to simplify the algebraic expression ( A) = 3q + q γ ( A) = 16q + 15q + 6q + q (8) For example, when σ = 15% and t = 0.5, then skewness is γ 1 (A) = 0.6 and excess kurtosis is γ (A) = Notice that skewness is always positive for the lognormal distribution. [Exhibit here.] Nonlognormal skewness and kurtosis for γ 1 (F) and γ (F) defined in equation (7) above give rise to implied volatility skews. To illustrate this effect, option prices are generated according to the JarrowRudd option price in equation (5) based on parameter values λ 1 = 0.5, λ = 5, S 0 = 450, σ = 0%, t = 3 months, r = 4%, and strike prices ranging from 400 to 500. Implied volatilities are then calculated for each skewness and kurtosisimpacted option price using the BlackScholes formula. The resulting volatility skew is plotted in Exhibit, where the horizontal axis measures strike prices and the vertical axis measures implied standard deviation values. While the true volatility value is σ = 0%, Exhibit reveals that implied volatility is greater than true volatility for deep outofthemoney options, but less than true volatility for deep inthemoney options. 6
8 [Exhibit 3 here.] Exhibit 3 contains an empirical volatility skew obtained from S&P 500 index call option price quotes recorded on December 1993 for options expiring in February In Exhibit 3, the horizontal axis measures option moneyness as the percentage difference between a dividendadjusted stock index level and a discounted strike price. Positive (negative) moneyness corresponds to inthemoney (outofthemoney) options with low (high) strike prices. The vertical axis measures implied standard deviation values. Solid marks represent implied volatilities calculated from observed option prices using the BlackScholes formula. Hollow marks represent implied volatilities calculated from observed option prices using the JarrowRudd formula. The JarrowRudd formula uses a single skewness parameter and a single kurtosis parameter across all price observations. The skewness parameter and the kurtosis parameter are estimated by a procedure described in the empirical results section below. There are actually 1,354 price quotes used to form this graph, but the number of visually distinguishable dots is smaller. Exhibit 3 reveals that BlackScholes implied volatilities range from about 17% for the deepest inthemoney options (positive moneyness) to about 9% for the deepest outofthemoney options (negative moneyness). By contrast, JarrowRudd implied volatilities are all close to 113% regardless of option moneyness. Comparing Exhibit 3 with Exhibit reveals that the BlackScholes implied volatility skew for these S&P 500 index options is consistent with negative skewness in the distribution of S&P 500 index prices. In the empirical results section of this paper, we assess the economic importance of these volatility skews. III. DATA SOURCES We base this study on the market for S&P 500 index options at the Chicago Board Options Exchange (CBOE), i.e., the SPX contracts. Rubinstein (1994) argues that this market best approximates conditions required for the BlackScholes model. Nevertheless, Jarrow and Rudd (198) point out that a stock index distribution is a convolution of its component distributions. Therefore, when the BlackScholes model is the correct model for individual stocks it is only an approximation for stock index options. 7
9 Intraday price data come from the Berkeley Options Data Base of CBOE options trading. S&P 500 index levels, strike prices and option maturities also come from the Berkeley data base. To avoid bidask bounce problems in transaction prices, we take option prices as midpoints of CBOE dealers bidask price quotations. The risk free interest rate is taken as the U.S. Treasury bill rate for a bill maturing closest to option contract expiration. Interest rate information is culled from the Wall Street Journal. Since S&P 500 index options are European style, we use Black s (1975) method to adjust index levels by subtracting present values of dividend payments made before each option s expiration date. Daily S&P 500 index dividends are collected from the S&P 500 Information Bulletin. Following data screening procedures in BaroneAdesi and Whaley (1986), we delete all option prices less than $0.15 and all transactions occurring before 9:00 A.M. Obvious outliers are also purged from the sample; including recorded option prices lying outside wellknown noarbitrage option price boundaries (Merton, 1973). IV. EMPIRICAL RESULTS In this section, we first assess the outofsample performance of the BlackScholes option pricing model without adjustments for skewness and kurtosis. Specifically, using option prices for all contracts within a given maturity series observed on a given day we estimate a single implied standard deviation using Whaley s (198) simultaneous equations procedure. We then use this implied volatility as an input to the BlackScholes formula to calculate theoretical option prices corresponding to all actual option prices within the same maturity series observed on the following day. Thus theoretical option prices for a given day are based on a priorday, outofsample implied standard deviation estimate. We then compare these theoretical prices with the actual market prices observed that day. Next, we assess the skewness and kurtosisadjusted BlackScholes option pricing formula developed by Jarrow and Rudd (198) using an analogous procedure. Specifically, on a given day we estimate a single implied standard deviation, a single skewness coefficient, and a single excess kurtosis coefficient using an expanded version of Whaley s (198) simultaneous equations 8
10 procedure. We then use these three parameter estimates as inputs to the JarrowRudd formula to calculate theoretical option prices corresponding to all option prices within the same maturity series observed on the following day. Thus these theoretical option prices for a given day are based on priorday, outofsample implied standard deviation, skewness, and excess kurtosis estimates. We then compare these theoretical prices with the actual market prices. The BlackScholes Option Pricing Model The BlackScholes formula specifies five inputs: a stock price, a strike price, a risk free interest rate, an option maturity and a return standard deviation. The first four inputs are directly observable market data. Since the return standard deviation is not directly observable, we estimate a return standard deviation implied by option prices using Whaley s (198) simultaneous equations procedure. This procedure yields a BlackScholes implied standard deviation (BSISD) that minimizes the following sum of squares. BSISD N [ OBS, j BS, j( )] min C C BSISD j = 1 (9) In equation (9) above, N denotes the number of price quotations available on a given day for a given maturity series, C OBS represents a marketobserved call price, and C BS (BSISD) specifies a theoretical BlackScholes call price based on the parameter BSISD. Using a priorday BSISD estimate, we calculate theoretical BlackScholes option prices for all contracts in a currentday sample within the same maturity series. We then compare these theoretical BlackScholes option prices with their corresponding marketobserved prices. [Exhibit 4 here.] Exhibit 4 summarizes results for S&P 500 index call option prices observed in December 1993 for options expiring in February To maintain exhibit compactness, column 1 lists only evennumbered dates within the month. Column lists the number of price quotations available on each of these dates. The BlackScholes implied standard deviation (BSISD) used to calculate theoretical prices for each date is reported in column 3. To assess differences between theoretical and observed prices, column 6 lists the proportion of theoretical BlackScholes option prices lying outside their corresponding bidask spreads, either below the bid price or above the asked price. 9
11 In addition, column 7 lists the average absolute deviation of theoretical prices from spread boundaries for those prices lying outside their bidask spreads. Specifically, for each theoretical option price lying outside its corresponding bidask spread, we compute an absolute deviation according to the following formula. max( C ( BSISD) Ask, Bid C ( BSISD) ) BS This absolute deviation statistic measures deviations of theoretical option prices from observed bidask spreads. Finally, column 4 lists daybyday averages of observed call prices and column 5 lists daybyday averages of observed bidask spreads. Since option contracts are indivisible, all prices are stated on a percontract basis, which for SPX options is 100 times a quoted price. In Exhibit 4, the bottom row lists column averages for all variables. For example, the average number of daily price observations is 1,18 (column ), with an average contract price of $,31.35 (column 4) and an average bidask spread of $56.75 (column 5). The average implied standard deviation is 1.88 percent (column 3). The average proportion of theoretical Black Scholes prices lying outside their corresponding bidask spreads is 75.1 percent (column 6), with an average deviation of $69.77 (column 7) for those observations lying outside a spread boundary. The average price deviation of $69.77 per contract for observations lying outside a spread boundary is slightly larger than the average bidask spread of $ However, price deviations are larger for deep inthemoney and deep outofthemoney options. For example, Exhibit 4 shows that the BlackScholes implied standard deviation (BSISD) estimated using Whaley s simultaneous equations procedure on December option prices is 15.9%, while Exhibit 3 reveals that contractspecific BlackScholes implied volatilities range from about 18% for deep inthemoney options to about 8% for deep outofthemoney options. Based on December SPX input values, i.e., S = $459.65, r = 3.15%, t = 78 days, a deep inthemoney option with a strike price of 430 yields call contract prices of $3, and $3,495.68, respectively, from volatility values of 18% and 15.9%. Similarly, a deep outofthemoney option with a strike price of 490 yields call contract prices of $46.0 and $395.13, respectively, from volatility values of 8% and 15.9%. These prices correspond to contract price deviations of $ for deep inthemoney options BS 10
12 and $ for deep outofthemoney options. These deviations are significantly larger than the average deviation of $56.75 per contract. Price deviations of the magnitude described above indicate that CBOE market makers quote deep inthemoney (outofthemoney) call option prices at a premium (discount) compared to BlackScholes prices. Nevertheless, the BlackScholes formula is a useful first approximation to deep inthemoney or deep outofthemoney option prices. Immediately below, we examine the improvement in pricing accuracy obtained by adding skewness and kurtosisadjustment terms to the BlackScholes formula. Skewness and KurtosisAdjusted JarrowRudd Model In the second set of estimation procedures, on a given day within a given option maturity series we simultaneously estimate a single return standard deviation, a single skewness parameter, and a single kurtosis parameter by minimizing the following sum of squares with respect to the arguments ISD, L 1, and L, respectively. ISD, L1, L N j= 1 [ OBS, j ( BS, j ( ) ) ] min C C ISD L Q L Q (10) The coefficients L 1 and L estimate the parameters λ 1 and λ, respectively, defined in equation (6), where the terms Q 3 and Q 4 are also defined. These daily estimates yield implied coefficients of skewness (ISK) and kurtosis (IKT) calculated as follows, where γ 1 (A) and γ (A) are defined in equation (7) above. ISK = L + γ ( A( ISD)) IKT = + L + γ ( A( ISD)) Thus ISK estimates the skewness parameter γ 1 (F) and IKT estimates the kurtosis parameter 3 + γ (F). Substituting estimates of ISD, L 1, and L into equation (5) yields skewness and kurtosisadjusted JarrowRudd option prices, i.e., C JR, expressed as the following sum of a Black Scholes option price plus adjustments for skewness and kurtosis deviations from lognormality. ( ) C = C ISD + L Q + L Q JR BS (11) Equation (11) yields theoretical skewness and kurtosisadjusted BlackScholes option prices from which we compute deviations of theoretical prices from marketobserved prices. 11
13 [Exhibit 5 here.] Exhibit 5 summarizes results for the same S&P 500 index call option prices used to compile Exhibit 4. Consequently, column 1 in Exhibit 5 lists the same evennumbered dates and column lists the same number of price quotations listed in Exhibit 4. The implied standard deviation (ISD), implied skewness coefficient (ISK), and implied kurtosis coefficient (IKT) used to calculate theoretical prices on each date are reported in columns 35. To assess the outofsample forecasting power of skewness and kurtosis adjustments, the implied standard deviation (ISD), implied skewness coefficient (ISK), and implied kurtosis coefficient (IKT) for each date are estimated from prices observed on the trading day immediately prior to each date listed in column 1. For example, the first row of Exhibit 5 lists the date December 1993, but columns 35 report that day s standard deviation, skewness and kurtosis estimates obtained from 1 December prices. Thus, outofsample parameters ISD, ISK and IKT reported in columns 35, respectively, correspond to oneday lagged estimates. We use these oneday lagged values of ISD, ISK and IKT to calculate theoretical skewness and kurtosisadjusted BlackScholes option prices according to equation (11) for all price observations on the evennumbered dates listed in column 1. In turn, these theoretical prices based on outofsample ISD, ISK and IKT values are then used to compute daily proportions of theoretical prices outside bidask spreads (column 6) and daily averages of deviations from spread boundaries (column 7). Like Exhibit 4, column averages are reported in the bottom row of the Exhibit 5. As shown in Exhibit 5, all daily skewness coefficients in column 4 are negative, with a column average of Daily kurtosis coefficients in column 5 have a column average of These optionimplied coefficients may be compared with sample coefficients reported in Exhibit 1 for the decade For example, optionimplied skewness of is comparable to logrelative return skewness of and arithmetic return skewness of calculated by including the October 1987 return. However, optionimplied kurtosis of 5.39 is less extreme than arithmetic return kurtosis of 9.7 and logrelative return kurtosis of 11.9 calculated by including the October 1987 return. This appears to suggest that any memory of the October 1987 crash embodied in S&P 500 option prices is more strongly manifested by negative optionimplied skewness than optionimplied excess kurtosis. 1
14 Column 6 of Exhibit 5 lists the proportion of skewness and kurtosisadjusted prices lying outside their corresponding bidask spread boundaries. The column average proportion is percent. Column 7 lists average absolute deviations of theoretical prices from bidask spread boundaries for only those prices lying outside their bidask spreads. The column average contract price deviation is $15.85, which is about onefourth the size of the average bidask spread of $69.77 reported in Exhibit 4. Moreover, Exhibit 3 reveals that implied volatilities from skewness and kurtosisadjusted option prices (hollow markers) are unrelated to option moneyness. In turn, this implies that the corresponding price deviations are also unrelated to option moneyness. Comparison of implied volatility values in Exhibits 4 and 5 suggests that the implied volatility series obtained using the JarrowRudd model is smoother than the series obtained using the BlackScholes model. Indeed, the average absolute value of daily changes in implied volatility is 0.4% for the JarrowRudd model versus a larger 0.91% for the BlackScholes model. Using a matchedpairs ttest on absolute values of daily changes in implied volatilities for both models we obtain a tvalue of 4.0 indicating a significantly smoother time series of implied volatilities from the JarrowRudd model. Thus not only does the JarrowRudd model flatten the implied volatility skew, it also produces more stable volatility estimates. The empirical results reported above will vary slightly depending on the assumed interest rate. For example, if the assumed rate is too low, implied standard deviation estimates will be biased upwards. Likewise, if the assumed rate is too high, implied volatility estimates will be biased downwards. In this study we follow standard research practice and use Treasury bill rates. However, these may understate the true cost of funds to option market participants. For example, Treasury bill repurchase (repo) rates better represent the true cost of borrowed funds to securities firms. For individual investors, however, the broker call money rate better represents a true cost of funds. To assess the robustness of our results to the assumed interest rate, we performed all empirical analyses leading to Exhibits 4 and 5 using Treasury bill repurchase rates and broker call money rates. On average, repurchase rates were 7 basis points higher than Treasury bill rates in December By contrast, call money rates were on average 196 basis points higher than Treasury bill rates. Implementing these interest rate alterations yielded the following results. First, average daily BlackScholes implied standard deviations are 1.81% using repurchase rates and 13
15 10.7% using call money rates. These are lower than the 1.88% average implied volatility reported in Exhibit 4. Second, for the JarrowRudd model using repurchase rates yields an average daily implied volatility of 11.55% and using call money rates yields an average volatility of 9.69%. Both are lower than the 11.6% average implied volatility reported in Exhibit 5. Also, using repurchase rates yields an average daily skewness coefficient of and an average daily kurtosis coefficient of But using call money rates yields an average daily skewness coefficient of and an average daily excess kurtosis coefficient of These are smaller in magnitude than the average skewness of and average kurtosis of 5.39 reported in Exhibit 5. But whichever interest rate is used to measure the cost of funds to S&P 500 options market participants, the optionimplied distributions of S&P 500 returns are still noticeably nonnormal. Overall, we conclude that skewness and kurtosisadjustment terms added to the Black Scholes formula yield significantly improved pricing accuracy for deep inthemoney or deep outofthemoney S&P 500 index options. Furthermore, these improvements are obtainable from outofsample estimates of skewness and kurtosis. Of course, there is an added cost in that two additional parameters must be estimated. But this cost is slight, since once the computer code is in place the additional computation time is trivial on modern computers. V. HEDGING IMPLICATIONS OF THE JARROWRUDD MODEL The JarrowRudd model has implications for hedging strategies using options. In this section, we derive formulas for an option s delta and gamma based on the JarrowRudd model. Delta is used to calculate the number of contracts needed to form an effective hedge based on options. Gamma states the sensitivity of a deltahedged position to stock price changes. By definition, delta is the first partial derivative of an option price with respect to the underlying stock price. Similarly, gamma is the second partial derivative. Taking first and second derivatives of the JarrowRudd call option price formula yields these delta and gamma formulas where the variables λ j and Q j were defined earlier in equation (6). C S Delta: N( d ) = 1 + λ1 Q3 + λ Q S S (1.a) 14
16 C Gamma: n ( d1)( S0 t ) S = σ + λ 1 Q λ 3 Q + S S (1.b) The first terms on the righthand side of the equations in (1) above are the delta and gamma for the BlackScholes model. Adding the second and third terms yields the delta and gamma for the JarrowRudd model. [Exhibit 6 here.] Exhibit 6 illustrates how a hedging strategy based on the JarrowRudd model might differ from a hedging strategy based on the BlackScholes model. In this example, S&P 500 index options are used to deltahedge a hypothetical $10 million stock portfolio with a beta of one. This example assumes an index level of S 0 = $700, an interest rate of r = 5%, a dividend yield of y = %, and a time until option expiration of t = 0.5. For the volatility parameter in the Black Scholes model, we use the average implied volatility of 1.88% reported in Exhibit 4. For the JarrowRudd model, we use the average implied volatility of 11.6% along with the average skewness and kurtosis values of λ 1 = and λ = 5.39 reported in Exhibit 5. In Exhibit 6, columns 1 and 4 list strike prices ranging from 660 to 750 in increments of 10. For each strike price, columns, 3, 5, and 6 list the number of S&P 500 index option contracts needed to deltahedge the assumed $10 million stock portfolio. Columns and 5 report the number of option contracts needed to deltahedge this portfolio based on the BlackScholes model. Columns 3 and 6 report the number of contracts needed to deltahedge based on the JarrowRudd model. In both cases, numbers of contracts required are computed as follows, where the option contract size is 100 times the index level (Hull, 1993). Number of contracts = Portfolio value / Contract size Option delta Exhibit 6 reveals that for inthemoney options a deltahedge based on the BlackScholes model specifies a greater number of contracts than a deltahedge based on the JarrowRudd model. But for outofthemoney options, a deltahedge based on the JarrowRudd model requires a greater number of contracts. Differences in the number of contracts specified by each 15
17 model are greatest for outofthemoney options. For example, in the case of a deltahedge based on options with a strike price of 740 the BlackScholes model specifies 601 contracts while the JarrowRudd model specifies 651 contracts. VI. SUMMARY AND CONCLUSION We have empirically tested an expanded version of the BlackScholes (1973) option pricing model developed by Jarrow and Rudd (198) that accounts for skewness and kurtosis deviations from lognormality in stock price distributions. The JarrowRudd model was applied to estimate coefficients of skewness and kurtosis implied by S&P 500 index option prices. Relative to a lognormal distribution, we find significant negative skewness and positive excess kurtosis in the optionimplied distribution of S&P 500 index prices. This observed negative skewness and positive excess kurtosis induces a volatility smile when the BlackScholes formula is used to calculate optionimplied volatilities across a range of strike prices. By adding skewness and kurtosisadjustment terms developed in the JarrowRudd model the volatility smile is effectively flattened. In summary, we conclude that skewness and kurtosisadjustment terms added to the BlackScholes formula yield significantly improved accuracy and consistency for pricing deep inthemoney and deep outofthemoney options. 16
18 REFERENCES BaroneAdesi, G. and R.E. Whaley (1986): The Valuation of American Call Options and the Expected ExDividend Stock Price Decline, Journal of Financial Economics, 17: Black, F. and Scholes, M. (1973): The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81: Black, F. (1975): Fact and Fantasy in the Use of Options, Financial Analysts Journal, 31:367. Hull, J.C. (1993): Options, Futures, and Other Derivative Securities, Englewood Cliffs, N.J.: Prentice Hall. Jarrow, R. and Rudd, A. (198): Approximate Option Valuation for Arbitrary Stochastic Processes, Journal of Financial Economics, 10: Merton, R.C. (1973): The Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, 4: Nattenburg, S. (1994): Option Volatility and Pricing, Chicago: Probus Publishing. Rubinstein, M. (1994): Implied Binomial Trees, Journal of Finance, 49: Stuart, A. and Ord, J.K. (1987): Kendall's Advanced Theory of Statistics, New York: Oxford University Press. Whaley, R.E. (198): Valuation of American Call Options on Dividend Paying Stocks, Journal of Financial Economics, 10:
19 EXHIBIT 1 HISTORICAL S&P 500 INDEX RETURN STATISTICS S&P 500 monthly return statistics for each of seven decades spanning 196 through Panel A statistics are based on arithmetic returns calculated as P t /P t11 and Panel B statistics are based on logrelative returns calculated as log(p t /P t1 ), where P t denotes an index value at the end of month t. Means and standard deviations are annualized. 95% confidence intervals for normal sample skewness and kurtosis coefficients are ± and 3 ± 0.877, respectively. A # indicates exclusion of the October 1987 crashmonth return. PANEL A: ARITHMETIC RETURNS Decade Mean Standard Deviation Skewness Kurtosis (%) (%) # PANEL B: LOGRELATIVE RETURNS Decade Mean Standard Deviation Skewness Kurtosis (%) (%) #
20 EXHIBIT 4 COMPARISON OF BLACKSCHOLES PRICES AND OBSERVED PRICES OF S&P 500 OPTIONS On each day indicated, a BlackScholes implied standard deviation (BSISD) is estimated from priorday option price observations. Currentday theoretical BlackScholes option prices are then calculated using this priorday volatility parameter estimate. All observations correspond to call options traded in December 1993 and expiring in February All prices are stated on a percontract basis, i.e., 100 times a quote price. Date Number of Price Observations Implied Standard Deviation (%) Average Call Price ($) Average BidAsk Spread ($) Proportion of Theoretical Prices Outside BidAsk Spreads (%) Average Deviation of Theoretical Price from Spread Boundaries ($) 1/0/ , /06/ , /08/ , /10/ , /14/ , /16/ , /0/ , // , /8/ /30/ , Average ,
21 EXHIBIT 5 COMPARISON OF SKEWNESS AND KURTOSIS ADJUSTED BLACKSCHOLES PRICES AND OBSERVED PRICES OF S&P 500 OPTIONS On each day indicated, implied standard deviation (ISD), skewness (ISK), and kurtosis (IKT) parameters are estimated from priorday price observations. Currentday theoretical option prices are then calculated using these outofsample parameter estimates. All observations correspond to call options traded in December 1993 and expiring in February All prices are stated on a percontract basis, i.e., 100 times a quote price. Date Number of Price Observations Implied Standard Deviation (%) Implied Skewness (ISK) Implied Kurtosis (IKT) Proportion of Theoretical Prices Outside BidAsk Spread (%) Average Deviation of Theoretical Prices from Spread Boundaries ($) 1/0/ /06/ /08/ /10/ /14/ /16/ /0/ // /8/ /30/ Average
22 EXHIBIT 6 NUMBER OF OPTION CONTRACTS NEEDED TO DELTAHEDGE A $10 MILLION STOCK PORTFOLIO Number of S&P 500 option contracts needed to deltahedge a $10 million stock portfolio with a beta of one using contracts of varying strike prices. This example assumes an index level of S o = $700, an interest rate of r = 5%, a dividend yield of y = %, a time until option expiration of t = 0.5. The BlackScholes (BS) model assumes a volatility of σ = 1.88% corresponding to the empirical value reported in Exhibit 4. The JarrowRudd (JR) model assumes σ = 11.6%, and skewness and kurtosis parameters of λ 1 = and λ = 5.39 corresponding to the empirical values reported in Exhibit 5. IntheMoney Options OutoftheMoney Options Strike Contracts (BS) Contracts (JR) Strike Contracts (BS) Contracts (JR)
23 EXHIBIT : IMPLIED VOLATILITY SKEW 3% % Implied Volatility 1% 0% 19% 18% Strike Price
24 EXHIBIT 3: IMPLIED VOLATILITIES (SPX: 1/0/93) 0% 18% Implied Volatility 16% 14% 1% 10% 8% 6% 4% % 0% % 4% 6% 8% Option Moneyness 3
25 EXHIBIT A.4 COMPARISON OF BLACKSCHOLES PRICES AND OBSERVED PRICES OF S&P 500 OPTIONS On each day indicated, a BlackScholes implied standard deviation (BSISD) is estimated from priorday option price observations. Currentday theoretical BlackScholes option prices are then calculated using this priorday volatility parameter estimate. All observations correspond to call options traded in December 1990 and expiring in January All prices are stated on a percontract basis, i.e., 100 times a quote price. Date Number of Price Observations Implied Standard Deviation (%) Average Call Price ($) Average BidAsk Spread ($) Proportion of Theoretical Prices Outside BidAsk Spreads (%) Average Deviation of Theoretical Price from Spread Boundaries ($) 90/1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ Average
26 EXHIBIT A.5 COMPARISON OF SKEWNESS AND KURTOSIS ADJUSTED BLACKSCHOLES PRICES AND OBSERVED PRICES OF S&P 500 OPTIONS On each day indicated, implied standard deviation (ISD), skewness (ISK), and kurtosis (IKT) parameters are estimated from priorday price observations. Currentday theoretical option prices are then calculated using these outofsample parameter estimates. All observations correspond to call options traded in December 1990 and expiring in January All prices are stated on a percontract basis, i.e., 100 times a quote price. Date Number of Price Observations Implied Standard Deviation (%) Implied Skewness (ISK) Implied Kurtosis (IKT) Proportion of Theoretical Prices Outside BidAsk Spread (%) Average Deviation of Theoretical Prices from Spread Boundaries ($) 90/1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ Average
27 EXHIBIT A.3: IMPLIED VOLATILITIES (SPX: 1/06/90) 35% 30% Implied Volatility 5% 0% 15% 10% 10% 5% 0% 5% 10% 15% Option Moneyness 6
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